Optimal Exploitation of Multiple
Stocks by a Common Fishery: A
New Methodology
Hilborn, R.
IIASA Research Report
August 1975
Hilborn, R. (1975) Optimal Exploitation of Multiple Stocks by a Common Fishery: A New Methodology. IIASA
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OPTIMAL EXPLOITATION OF MULTIPLE STOCKS
BY A COMMON FISHERY: A NEW METHODOLOGY
Ray Hilborn
August 1975
Research Reports are publications reporting
on the work of the author. Any views or
conclusions are those of the author, and do
not necessarily reflect those of IIASA.
Optimal Exploitation Of Multiple Stocks
By A Colmon Fishery: A New Methodology
Ray Hilborn*
Abstract
Optimal harvest rates for mixed stocks of fish are
calculated using stochastic dynamic programming. This
technique is shown to be superior to the best methods
currently described in the literature. The Ricker stock
recruitment curve is assumed for two stocks harvested
by the same fishery. The optimal harvest rates are
calculated as a function of the size of each stock, for
a series of possible parameter values. The dynamic programming solution is similar to the fixed escapement
policy only when the two stocks have similar Ricker parameters, or when the two stocks are of equal size.
Normally, one should harvest harder than calculated from
fixed escapement analysis.
Introduction
It is well recognized that many fisheries exploit more
than one stock of fish: a stock may consist of separate
species at .~arioustrophic levels, as in tropical fisheries,
or genetically isolated races of the same species, as in Pacific
salmon. The problem of optimal harvesting of these mixed fisheries is interesting because the biological understanding of
the stock dynamics is frequently quite advanced relative to the
methodological tools to de~erminethe optimal harvest. Paulik
et al. [ 7 ] have presented techniques for calculating the optimal
harvest rate for fisheries consisting of up to twenty separate
stocks. They use the basic Ricker equation of stock dynamics
(Ricker [9] which assumes a deterministic relationship between
spawning stock and resultant run. Their solution involves
solving a set of equations iteratively by computer to arrive at
the optimal exploitation rate.
* Institute of
Animal Resource Ecology, University of
British Columbia, Nancouver, B.C., Canada, and the International
Institute for Applied Systems Analysis, Laxenburg, Austria.
There are several weaknesses in their solution, which are
due to the analytic intractability of the problem. The authors
calculate only the optimal exploitation rate, assuming the
population is at equilibrium. There is incredible variation in
the actual stock recruitment relationships for salmon populations,
and current management practice uses the concept of fixed
escapement instead of fixed harvest rate. The fixed escapement policy recognizes that the long term yield is maximized by
allowing a fixed number of adult salmon to reach the spawning
grounds, irrespective of the number of salmon in the total run.
When the stock is at high numbers it can be harvested at a
higher rate than when it is at low numbers. Ricker [lo] has
calculated optimal escapement for several stock recruitment relationships using numerical methods. Paulik et al. calculated
the total run and optimal exploitation rate of all stocks at
equilibrium. To derive the escapement one multiplies total run
times the exploitation rate. It is not clear, however, that a
fixed escapement policy is optimal for mixed fisheries. It
is shown later in this paper that the optimal escapement is not
independent of the relative abundances of the different stocks.
Specifically, if the fishery consists of two stocks, determination of the optimal expl::.itation rate, or escapement, will
depend on the sizes of the two stocks. This is not just a
theoretical possibility; data collection associated with current management of salmon provides reasonably accurate run
estimates of stock sizes so that it is definitely possible to
implement these policies.
Methods
Current methods for determination of optimal exploitation
rates use simple analytic analysis of very simple stock recruitment models to determine optimal exploitation rates at
equilibrium. Much more complicated computer simulation models
have been used to study fish stock dynamics, (Larkin and
Hourston [5]; Ward and Larkin [15])--but these models have not
been used to determine optimal exploitation rates. It is
possible to use more complex models to test very simple control
laws; for instance, constant harvest or constant escapement.
You simply have the same harvest taken every year and then
calculate the average catch by simulating a large number of
years. This method has been used to look at the role of
stochastic variation on simple stock recruitment models
(Ricker [I01 , Larkin and Hourston [5])
It is theoretically
possible and computationally practical to do the same sort of
analysis on very complex models (Peterman [81). The main
limitation is that the harvest policy must be the same every
year. The harvest policy cannot be tied to the size of the
various stocks except by fixing a total escapement. If we try
to use a simulation approach for every possible combination of
harvest rates as a function of stock sizes, the number of computations required rapidly exceeds the ability of modern digital
.
computers. It is easy to understand that we wish to harvest
harder when a stock is high than when it is low, as the fixed
escapement policy automatically does for a single stock. But
for a two-stock example, a fixed escapement policy does not
differentiate between a case where two stocks are at moderate
densities, and a case where one stock is very low and one is
very high. The long term harvest can be increased by determining the harvest rate as a function of the stock sizes of both
stocks, when harvesting a mixed stock.
A new methodology has recently been introduced to fisheries
management (Walters [I41 which eliminates the computational
constraints and greatly widens the scope of optimization in
fisheries. Walters used the technique of stochastic dynamic
programing first developed by Bellman. (See Bellman [I];
Bellman and Dreyfus [2]; Bellman and Kalaba [3]. For other
applications of dynamic programming to ecological problems, see
Shoemaker [I31 , and Sancho [ I 1 1 .) For a good description of
stochastic dynamic programming, see Walters [141. Briefly,
stochastic dynamic programming allows one to calculate optimal
control policies by a procedure that involves the number of
computations increasing linearly, instead of geometrically, with
the number of time steps. It requires approximation due to
discretization of the state vari.ables (stock size) and the control policies (harvest rates). Walters used an example of a
single salmon stock, discretized into thirty population levels,
with thirty discretized exploitation rates and ten discrete
stochastic possibilities. This requires running a simulation of
the stock dynamics 9 0 0 0 times per year. Using the simulation
approach of following all possible paths into the future, say
20
twenty years, this would have required 9 0 0 0
simulations,
clearly beyond the scope of current computers. However, using
stochastic dynamic programming, only 9 0 0 0 x 2 0 simulations were
required. This requires only a few seconds on a modern digital
computer.
stochastic dynamic programming has five main advantages
over previous analytic techniques. They are:
1)
The stock recruitment model can be as complex as
desired; the number of parameters in the model does
not affect the computation time required or the reliability of the results.
2)
Parameters may be stochastic. However, as the number
of stochastic possibilities considered for the parameter values increase, so does computation time.
3)
There may be judgmental uncertainty about parametric
values. This is analogous to the stochastic variability of parameters, but conceptually distinct.
4)
The objective function (what is maximized) can be
as complex as desired. It does not need to be
" l o n g t e r m c a t c h " ; it c a n b e " d o l l a r v a l u e o f c a t c h , "
" t o t a l employment g e n e r a t e d from t h e f i s h e r y , " o r a n y
combination of f a c t o r s .
5)
D i s c o u n t i n g r a t e s c a n b e i n t r o d u c e d i n t o t h e model
w i t h no problem.
The t o t a l o b j e c t i v e d o e s n o t need t o
b u t may b e m u l t i p l i c a t i v e
b e summed o v e r t i m e [ C ( O i ) ] ,
[nu
+ oil].
Although t h e number o f c o m p u t a t i o n s g o e s up l i n e a r l y w i t h
t h e number o f t i m e i n t e r v a l s , i t g o e s u p g e o m e t r i c a l l y w i t h
t h e number o f s t a t e v a r i a b l e s and s t o c h a s t i c p a r a m e t e r s .
Thus
w e a r e p r a c t i c a l l y r e s t r a i n e d t o o p t i m i z i n g models w i t h on t h e
o r d e r of f i v e s t a t e v a r i a b l e s .
I have chosen t o u s e t h e s t a n d a r d Ricker s t o c k r e c r u i t m e n t
model o f salmon dynamics ( R i c k e r [ 9 ] ) . Most w i l l remember:
R =
s
exp ( a ( l
-
g))
,
(1
where
R = t h e t o t a l number o f o f f s p r i n g t h a t w i l l r e t u r n a s a d u l t s ,
S = t h e number o f s p a w n e r s ,
a = a parameter of p r o d u c t i v i t y ,
B = t h e number o f spawners a t which t h e a v e r a g e number o f
r e t u r n i n g f i s h p e r spawner i s o n e .
I h a v e c h o s e n t h i s model b e c a u s e it h a s b e e n u s e d by a l m o s t
a l l r e c e n t work o n salmon s t o c k dynamics, and p a r t i c u l a r l y by
P a u l i k e t a l . [ 7 ] , and W a l t e r s [ 1 4 ] .
his f a c i l i t a t e s comI used twenty d i s c r e t e l e v e l s f o r each s t o c k
parison of r e s u l t s .
o f e i g h t e e n d i s c r e t e h a r v e s t r a t e s , and t e n s t o c h a s t i c o u t comes. A l t h o u g h a R i c k e r model was u s e d f o r t h e s t o c k r e c r u i t ment r e l a t i o n s h i p , o t h e r commonly u s e d models o f f i s h s t o c k
r e c r u i t m e n t s u c h a s t h e B e v e r t o n - H o l t ( B e v e r t o n and H o l t [ 4 1 )
o r t h e S c h a e f e r model ( S c h a e f e r [ I 2 1 ) c o u l d b e s u b s t i t u t e d .
The s t a t e o f a s i n g l e s t o c k a t a t i m e i n t e r v a l i s d e s c r i b e d
by a s i n g l e number, t h e s t o c k s i z e . W e c a n i n t h e o r y d e a l w i t h
up t o a b o u t f i v e s e p a r a t e s t o c k s w i t h o u t r u n n i n g i n t o comput a t i o n a l problems.
However, it i s d i f f i c u l t t o p r e s e n t and
understand t h e r e s u l t s of optimization w i t h f i v e s t a t e v a r i a b l e s , s o I have c h o s e n t o u s e j u s t two s t o c k s f o r d e m o n s t r a t i o n
purposes.
I f t h i s t e c h n i q u e w e r e u s e d i n a c t u a l management;
it c o u l d e a s i l y b e used on mixed s t o c k s o f f i v e s e p a r a t e s t o c k s .
Results
S i n c e t h e c a l c u l a t i o n of o p t i m a l c o n t r o l p o l i c i e s r e q u i r e s
c o m p u t a t i o n s on a computer, no g e n e r a l s o l u t i o n c a n be p r e s e n t e d . What I w i l l do i s p r e s e n t o p t i m a l c o n t r o l s o l u t i o n s
f o r a s e r i e s of p o s s i b l e parameter v a l u e s f o r two s t o c k s , and
g e n e r a l i z e from t h e s e r e s u l t s .
From e q u a t i o n ( 1 ) we c a n s e e
t h a t t h e dynamics of e a c h s t o c k a r e governed by two p a r a m e t e r s ,
a and B.
For any two s t o c k s , t h e r e a r e f i v e u n i q u e r e l a t i o n s h i p s between p a r a m e t e r s .
They a r e :
1)
a
2)
one s t o c k h a s a h i g h e r a v a l u e , and B v a l u e s a r e t h e
same;
3)
s t o c k 1 h a s a lower a v a l u e , and s t o c k 2 h a s a lower
B value;
4)
s t o c k 1 h a s a lower a v a l u e and a lower B v a l u e ;
5)
t h e a v a l u e s a r e t h e same, b u t one h a s a lower B v a l u e .
v a l u e s a r e t h e same and B v a l u e s a r e t h e same;
S t o c h a s t i c dynamic programming c a l c u l a t e s a c o n t r o l law
( h a r v e s t r a t e ) a s a f u n c t i o n of t h e s t a t e v a r i a b l e s ( t h e two
r u n s i z e s ) . To p r e s e n t t h e c o n t r o l laws g e n e r a t e d by t h e o p t i m i z a t i o n p r o c e d u r e , I drew h a r v e s t r a t e i s o c l i n e s on a g r i d
w i t h t h e r u n s i z e of s t o c k 1 on t h e X-axis and t h e r u n s i z e of
F i g u r e 1 p r e s e n t s t h e c o n t r o l laws
s t o c k 2 on t h e Y-axis.
f o r a c a s e where s t o c k 1 h a s a n a v a l u e of 1.0 and a B v a l u e of
1.0.
S t o c k 2 h a s a n a v a l u e of 2.2 and a B v a l u e of 0.4.
These p a r a m e t e r s c o r r e s p o n d t o c a s e 3 above.
The i s o c l i n e s f o r h a r v e s t r a t e s of 0, 0.3, 0.5, and 0.7
a r e drawn.
S i n c e s t o c k 2 , on t h e Y-axis, i s more p r o d u c t i v e ,
t h e r e i s a h i g h e r h a r v e s t r a t e f o r Low v a l u e s of s t o c k 2 t h a n
t h e r e i s f o r low v a l u e s of s t o c k 1 .
I n o r d e r t o compare
t h e s e r e s u l t s w i t h a c o n s t a n t escapement p o l i c y , we must u t i l i z e some s i m p l e r e l a t i o n s h i p s . We know t h a t :
Escapement = (Run of s t o c k 1 + Run of s t o c k 2)
* (Harvest r a t e ) .
(2)
From t h i s we c a n c a l c u l a t e t h a t
Run of s t o c k 2 =
Escapement
H a r v e s t Rate
-
Run of s t o c k 1
. (3)
This equation enables u s t o p l o t t h e harvest r a t e i s o c l i n e s
on t h e s t o c k 1 , s t o c k 2 s u r f a c e .
It is also evident t h a t a l l
STOCK 1
Figure 1.
Harvest r a t e i s o c l i n e s derived
from dynamic programming ( t h i c k
s o l i d l i n e ) and f i x e d escapement
a and B v a l u e s
(dashed l i n e ) .
a r e 1.0 and 1.0 f o r s t o c k 1 ,
a n d 2 . 2 arid 1 . 0 f o r s t o c k 2.
i s o c l i n e s w i l l h a v e a s l o p e of -1.
T h i s means t h a t u s i n g a
c o n s t a n t escapement p o l i c y , t h e o p t i m a l h a r v e s t r a t e l i n e s w i l l
a l w a y s b e t h e same s h a p e , i n d e p e n d e n t of t h e p a r a m e t e r s a and
B.
The h a r v e s t r a t e i s o c l i n e s under a f i x e d escapement p o l i c y
have been drawn a s d a ~ h e dl i n e s i n F i g . 1 . I t i s o b v i o u s t h a t
t h e o p t i m a l s o l u t i o n from t h e dynamic programming a l g o r i t h m i s
q u i t e d i f f e r e n t from t h e f i x e d escapement law d e r i v e d from
P a u l i k e t a l . [ 7 ] . F i g . ~ ~ r e2 s and 5 p r e s e n t s i m i l a r p l o t s f o r
c a s e s 1 , 2 , 4 and 5.
Discussion
From t h e r e s u l t s i n F i g u r e s 1 - 5, i t i s c l e a r t h a t f i x e d
escapement i s t h e o p t i m a l p o l i c y o n l y when t h e two s t o c k s
have t h e same a and B v a l u e s ( c a s e 1 ) . Thus, a f i x e d e s c a p e ment p o l i c y f o r managing mixed s t o c k s of salmon i s o p t i m a l
o n l y under very r e s t r i c t i v e circumstances.
The r e s u l t s o b t a i n e d
above s u g g e s t t h a t i n g e n e r a l one s h o u l d h a r v e s t a mixed s t o c k
h a r d e r when t h e r a t i o o f t h e two s t o c k s s t r a y s away from 1 : l .
T h i s s u g g e s t s t h a t a s o n e s t o c k becomes much more s i g n i f i c a n t t h a n t h e o t h e r , t h e management s h o u l d p r o c e e d a s i f i t
w e r e t h e only stock.
I t appears t h a t t h e expected b e n e f i t s
from r e d u c i n g t h e h a r v e s t r a t e s when o n e s t o c k becomes low
a r e outweighed by t h e l o s s o f c a t c h from t h e r e d u c e d h a r v e s t .
I t must b e s t r e s s e d however, t h a t t h e s e c o n c l u s i o n s a p p l y o n l y
f o r t h e o b j e c t i v e f u n c t i o n maximized:
expected annual average
yield.
I f o t h e r f a c t o r s s u c h a s s t o c k d i v e r s i t y w e r e t o be
i n c l u d e d i n t h e o b j e c t i v e f u n c t i o n , t h e o p t i m a l c o n t r o l laws
would u n d o u b t e d l y c h a n g e .
S t o c h a s t i c dynamic programminq a p p e a r s t o be t h e b e s t
c u r r e n t method f o r p r o d u c i n g c o n t r o l l a w s f o r mixed s t o c k s o f
fishes.
The f a c t t h a t t h e above examples w e r e worked f o r
P a c i f i c salmon s h o u l d n o t c a u s e o n e t o f o r g e t t h a t t h e t e c h niques used a r e completely g e n e r a l i z a b l e t o a very l a r g e c l a s s
The p r i m a r y l i m i o f f i s h e r i e s and o t h e r e c o l o g i c a l problems.
t a t i o n i s i n t h e number o f s t a t e v a r i a b l e s , b u t f o r any renewable r e s o u r c e where some a n a l o g o f a s t o c k r e c r u i t m e n t c u r v e
can be c o n s t r u c t e d , t h e n a s i n g i e v a r i a b l e , t h e s t o c k , i s suff i c i e n t t o d e s c r i b e t h e p o p u l a t i o n , and s t o c h a s t i c dynamic
programming c a n be u s e d .
The n a i n l i m i t a t i o n s o c c u r when a g e /
c l a s s phenomena become i m p o r t a n t , s o t h a t s e v e r a l numbers a r e
required t o describe a population.
However, f o r a l m o s t a l l
f i s h e r i e s problems, a stock-recruitment r e l a t i o n s h i p i s t h e
b a s i s o f p r e s e n t management, s o u s i n g dynamic programming a s a n
(See
o p t i m i z a t i o n t e c h n i q u e would s e e m t o be most a p p r o p r i a t e .
Parrish [6] )
.
Figure 2.
Harvest r a t e i s o c l i n e s f o r c a s e
1 in text.
S o l u t i o n from
dynamic programming a n d f i x e d
escapement a r e i d e n t i c a l . a a n d
B values f o r both stocks a r e 1.8
and 1 . O .
STOCK 1
Figure 3 .
Harvest r a t e i s o c l i n e s derived
from dynamic programming ( t h i c k
s o l i d l i n e s ) and f i x e d escapement ( d a s h e d l i n e s ) . a and B
v a l u e s a r e 1.0 and 1.0 f o r s t o c k
1 , and 2 . 2 and 1 . 0 f o r s t o c k 2 .
STOCK 1
Figure 4.
Harvest r a t e i s o c l i n e s derived
f r o m d y n a m i c programming ( t h i c k
s o l i d l i n e s ) and f i x e d e s c a p e ment ( d a s h e d l i n e s ) . a a n d B
v a l u e s a r e 1.0 and . 4 f o r s t o c k
1, a n d 2 . 2 a n d 1 . 0 f o r s t o c k 2 .
F i g u r e 5.
Harvest r a t e i s o c l i n e s derived
f r o m d y n a m i c programming ( t h i c k
s o l i d l i n e s ) and f i x e d escapea a n d El
ment ( d a s h e d l i n e s )
v a l u e s a r e 1 . 8 and . 4 f o r s t o c k
1 , a n d 1 . 8 and 1 . 0 f o r s t o c k 2 .
.
T a b l e 1.
a and B v a l u e s u s e d i n o p t i m i z a t i o n s .
Case No.
Stock 2
Stock 1
a
B
a
B
1
1.8
1 .O
1.8
1 .O
2
1.0
1.0
2.2
1.0
3
1 .O
1 .O
2.2
0.4
4
1.0
0.4
2.2
1 .O
5
1.8
0.4
1.8
1 .O
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